coffee break 8 (probability measures)

It is well known that one of the main ideas underlying in Black-Scholes’ formula is the risk neutral probability.  

As the risk neutral probability might be gotten by solving given equations, it is usually explained in Finance that the risk neutral probability is artificial.

However, in mathematics, every probability measures $P$ is artificial. Because those may just satisfy the only following definitions.

(1)$P(\Omega)=1,\quad P(\phi)=0$

(2)$0\leq P(A) \leq 1\quad (A\subset \Omega)$
(3)$P(\cup A_i)=\sum P(A_i)$

Such the axiomatic probability theory was founded by Kolmogorov in the 1930s.

Many mathematicians accepted gladly the new paradigm.

Then, the axiomatic structure based upon Lebesgue theory gave birth to big progress in the probability theory.

However, on the other hand, as Kiyoshi Ito has once noted, the structure has made an enormous gap between those who are familiar with mathematics and those who are not.